There has been much recent interest in random graphs sampled uniformly from the n-vertex graphs in a suitable minor-closed class, such as the class of all planar graphs. We shall recall some background, and then use combinatorial and probabilistic methods to extend these results.
We will consider random graphs from a `well-behaved’ class of graphs: examples of such classes include all minor-closed classes of graphs with 2-connected excluded minors (such as forests, series-parallel graphs and planar graphs), the class of graphs embeddable on any given surface, and the class of graphs with at most k vertex-disjoint cycles. Also, we will give weights to edges and components to specify probabilities, so that our random graphs correspond to the “random cluster” model, appropriately conditioned.
We find that earlier results extend naturally in both directions, to general well-behaved classes of graphs, and to the weighted framework, for example results concerning the probability of a random graph being connected, and we also find new results on the 2-core.